The evolution of the topological structure of the integral quantum Hall effect

The topological structure of the integral Hall conductance is determined by making use of the decomposition of the gauge potential on a U(1) principal fibre bundle over k-space. The integral Hall conductance density is found to take the form of a delta-function delta(psi). Only the zeroes of Bloch waves contribute to the Hall conductance. The topological structure of the integral Hall conductance is labeled by the Hopf index beta and the Brouwer degree eta. The Hall conductance topological current and its evolution is discussed. The Hall topological charges are found generating or annihilating at the limit points of wave function and splitting and merging at the bifurcation points of the wave function. Different directions of the bifurcation are given. (C) 1998 Published by Elsevier Science B.V.